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Math Help - Prove the following are equivalent

  1. #1
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    Prove the following are equivalent

    Let R be a nonzero ring. Prove that the following are equivalent:

    a)R is a field
    b)The only ideals in R are (0) and (1).
    c)Every homomorphism of R into a nonzero ring S is injective.
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  2. #2
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    Quote Originally Posted by apple2010 View Post
    Let R be a nonzero ring. Prove that the following are equivalent:

    a) R is a field
    b) The only ideals in R are (0) and (1).
    c) Every homomorphism of R into a nonzero ring S is injective.
    you forgot to mention that R is commutative.

    a) --> b) is trivial by the definition of a field. for b) --> c) look at the kernel of the homomorphism, which is an ideal of R.

    for c) --> a) suppose that r is a non-zero element of R and put I = Rr, S = R/I and take f : R ---> S to be the natural homomorphism.
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